The Gårding cones in the modern theory of fully nonlinear second order differential equations

Ivochkina, N. M.; Prokof’eva, S. I.; Yakunina, G. V.

doi:10.1007/s10958-012-0869-1

Cited by 15 publications

(17 citation statements)

References 26 publications

(46 reference statements)

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“…For the Gårding cone in the space of symmetric matrices, similar results to (4.11) can be seen Section 3 in [9].…”

Section: Appendixsupporting

confidence: 70%

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Local solvability of the k-Hessian equations

Tian

Wang

2016

Sci. China Math.

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Abstract. In this work, we study the existence of local solutions in R n to k-Hessian equation, for which the nonhomogeneous term f is permitted to change the sign or be non negative; if f is C ∞ , so is the local solution. We also give a classification for the second order polynomial solutions to the k−Hessian equation, it is the basis to construct the local solutions and obtain the uniform ellipticity of the linearized operators at such constructed local solutions.

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“…For the Gårding cone in the space of symmetric matrices, similar results to (4.11) can be seen Section 3 in [9].…”

Section: Appendixsupporting

confidence: 70%

“…In this paper, we always discuss the k-Hessian equation under the framework of ellipticity, then we follow the ideas of [8] and [9] to get the existence of the solution. Here we explain the ellipticity: in the matrix language, the ellipticity set of the k-Hessian operator, 1 ≤ k ≤ n, is…”

Section: Introductionmentioning

confidence: 99%

Local solvability of the k-Hessian equations

Tian

Wang

2016

Sci. China Math.

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“…The significance of L.Gårding theory in FNPDE was firstly revealed in the paper [2], p.268, Section 1. The paper [22] contains a description of algebraic aspects of L.Gårding theory and renovated proofs of basic theorems. The paper [7] contains a popular review of L.Gårding theory and its relations to T m , K m and FNPDE.…”

Section: Cone Of M-positive Matricesmentioning

confidence: 99%

On two symmetries in the theory of $m$-Hessian operators

Ivochkina¹,

Filimonenkova²

2018

TMNA

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The paper introduces a new differential-geometric system which originates from the theory of m-Hessian operators. The core of this system is a new notion of invariant differentiation on surfaces. This novelty gives rise to the following absolute geometric invariants: invariant derivatives of the surface position vector, an invariant connection on a surface via subsurface, curvature matrices of a hypersurface and its normal sections, p-curvatures and m-convexity of a hypersurface, etc. Our system also produces a new interpretation of the classic geometric invariants and offers new tools to solve geometric problems. In order to expose an application of renovated geometry we deduce an a priori C 1 -estimate for solutions to the Dirichlet problem for m-Hessian equations.

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“…The development of the theory of Hessian equations has brought out some new notions in differential geometry and the first description of some may be found in [1] as necessary conditions for admissible solvability of the Dirichlet problems. In the papers [13], [10], [6] some versions of these requirements were considered independently of differential equations as the set of new geometric notions. Namely, let ∂Ω ∈ R n be C 2 -hypersurface with positionvector X = X(θ) and metric tensor g[∂Ω] = (g ij ) n−1…”

Section: Notations and Definitionsmentioning

confidence: 99%